A note on Atiyah's \(\Gamma\)-index theorem in Heisenberg calculus (Q1689865)

Let \(\tilde{M}\to M\) be a covering of a compact manifold. Differential operators on the base \(M\) correspond to differential operators on \(\tilde{M}\) that are invariant under deck transformations. For an elliptic differential operator \(D\) on \(M\), Atiyah defines an \(L^2\)-index for the corresponding differential operator on \(\tilde{M}\) and shows that it is equal to the usual index. Here ellipticity is only used to define the relevant indices. When one replaces the ordinary pseudodifferential calculus by another one, one gets a different notion of ellipticity. Atiyah's \(L^2\)-index theorem and its proof carry over to this new setting. This note works this out for a differential operator that is elliptic in the Heisenberg calculus, defining an \(L^2\)-index for the corresponding differential operator on the covering and showing that it is equal to the usual index. It computes the index explicitly in an example where \(M\) is the \(3\)-torus equipped with a certain contact manifold structure.